Introduction EM-TV Methods Numerical Results Bregman-EM-TV Methods for Deconvolution Problems with Poisson Noise Christoph Brune, Alex Sawatzky Prof. Martin Burger Institute of Computational and Applied Mathematics University of M¨ unster, Germany Conference on Applied Inverse Problems, Vienna 2009 1 Christoph Brune University of M¨ unster
Introduction EM-TV Methods Numerical Results Outline Introduction 1 Inverse Problems in Microscopy Goal / Motivation EM-TV Methods 2 EM / Richardson-Lucy Nested EM-TV and Analysis Contrast Enhancement by Bregman-EM-TV Numerical Results 3 2 Christoph Brune University of M¨ unster
Introduction Inverse Problems in Microscopy EM-TV Methods Goal / Motivation Numerical Results STED-Microscope St imulated E mission D epletion (Prof. Hell, MPI G¨ ottingen) Fluorescence by focused laser beam; Stimulation + Depletion 3 Christoph Brune University of M¨ unster
Introduction Inverse Problems in Microscopy EM-TV Methods Goal / Motivation Numerical Results Optical Nanoscopy in so far unknown sharpness STED microscope overcomes physical limits imposed by diffraction Correlation between laser intensity, convolution kernel, granularity Industry Partner: Leica Microsystems CMS GmbH 4 Christoph Brune University of M¨ unster
Introduction Inverse Problems in Microscopy EM-TV Methods Goal / Motivation Numerical Results Image Reconstruction Basic problem: We consider the inverse problem, a linear equation K ˜ u + b = g K : L 1 (Ω) → L 1 (Σ) linear and compact operator with non-closed range and obtaining positivity � ill-posed problem (Hadamard) u exact image (desired solution) ˜ b assumed constant background model g exact data (values before detection) Typical examples are Fredholm integral operators of the first kind � ( Ku )( x ) = k ( x , y ) u ( y ) dy , x ∈ Σ , k non-negative . Ω 5 Christoph Brune University of M¨ unster
Introduction Inverse Problems in Microscopy EM-TV Methods Goal / Motivation Numerical Results Reconstruction in Optical Nanoscopy Deconvolution problems with Poisson noise � Ku + b = f ( Ku )( x ) = h ( x − τ ) u ( τ ) d τ Ω Nanoscopy provides blurred images Statistical noise model, described by likelihood p ( f | u ) → f i realization of random var. F i Poisson-noise resulting from laser sampling (photon counts) exact data ˜ u kernel h (psf) convolved data convolved & noisy data f 6 Christoph Brune University of M¨ unster
Introduction Inverse Problems in Microscopy EM-TV Methods Goal / Motivation Numerical Results Reconstruction in Optical Nanoscopy Deconvolution problems with Poisson noise � Ku + b = f ( Ku )( x ) = h ( x − τ ) u ( τ ) d τ Ω Nanoscopy provides blurred images Statistical noise model, described by likelihood p ( f | u ) → f i realization of random var. F i Poisson-noise resulting from laser sampling (photon counts) exact data ˜ u kernel h (psf) convolved data convolved & noisy data f 6 Christoph Brune University of M¨ unster
Introduction Inverse Problems in Microscopy EM-TV Methods Goal / Motivation Numerical Results Goal / Motivation 3D image reconstruction method Accurate, fast and robust deblurring and noise removal Cartoon reconstructions → segmentation, quantification Well-posedness of a continuous variational model Positivity preserving algorithm Proved convergence of algorithm Simultaneous contrast enhancement 7 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV Expectation Maximization (EM) Statistical modeling, Bayes theory Formulation as a random experiment F i = Poiss { ( Ku ) i + b } EM-Algorithm by Dempster, Laird, Rubin (E)step: Averaging complete data log-likelihood over its cond. distr. (M)step: Maximization of conditional expectation Use of maximum a-posteriori (MAP) estimation wanted: ˆ u = arg max p ( u | f ) u 8 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV MAP Likelihood Approach Consider the a-posteriori likelihood: p ( f | u ) p ( u ) p ( u ) ∼ e − α R ( u ) ( Gibbs ) p ( u | f ) = , p ( f ) � i . i . d . = p λ ( F = f ) · p ( u ) , λ = E [ F ] = Ku x ∈ Σ Ku ( x ) f ( x ) � e − Ku ( x ) · e − α R ( u ) = f ( x )! x ∈ Σ = ⇒ ˆ u = arg max p ( u | f ) = arg min u {− log p ( u | f ) } u � = arg min u { Ku − f log Ku + f log f − f + α R ( u ) } � �� � Σ independent of u Kullback-Leibler divergence Regularization 9 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV Variational Formulation for R ( u ) = 0 � min Ku − f log Ku u Σ Optimality condition � f � K ∗ 1 − K ∗ (Scal. K ∗ 1 = 1) = 0 , Ku � f � u = u K ∗ Leads to fixed point equation Ku � f � 2 = u k K ∗ Iteration scheme (EM step) u k + 1 Ku k 10 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV Variational Formulation for R ( u ) = 0 � min Ku − f log Ku u ≥ 0 Σ Optimality condition / KKT : ∃ λ ≥ 0 � f � K ∗ 1 − K ∗ (Scal. K ∗ 1 = 1) λ u = 0 , − λ = 0 , Ku � f � u = u K ∗ Leads to fixed point equation Ku � f � 2 = u k K ∗ Iteration scheme u k + 1 (EM step) Ku k 10 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV Variational Formulation for R ( u ) = 0 � min Ku − f log Ku u ≥ 0 Σ Optimality condition / KKT : ∃ λ ≥ 0 � f � K ∗ 1 − K ∗ (Scal. K ∗ 1 = 1) λ u = 0 , − λ = 0 , Ku � f � u = u K ∗ Leads to fixed point equation Ku � f � 2 = u k K ∗ Iteration scheme u k + 1 (EM step) Ku k Convergence analysis for exact data: descent in objective functional 10 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV Example: Nanoscopy EM, KL-Distance: 3 . 20 Convolved + Poisson noise 11 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV Motivation for TV Regularization Need additional (a-priori) information about: Known structures in the image Desired structures to be investigated EM algorithm uses uniform prior probability distribution Prior probability can be related to regularization functional p ( u ) ∼ e − α | u | BV TV-Regularization to preserve edges � � R ( u ) := | u | BV = sup u div g ( = |∇ u | ) g ∈ C ∞ 0 (Ω; R d ) Ω Ω || g || ∞ ≤ 1 12 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV EM-Algorithm and TV-Regularization � min Ku − f log Ku + α | u | BV , α > 0 u ∈ BV (Ω) Σ Combining nonlocal part (incl. K ) and local regularization functional Convex, but not differentiable Optimality condition for u > 0 K ∗ 1 − K ∗ ( f Ku ) + α p = 0 , p ∈ ∂ | u | BV K ∗ 1 − K ∗ ( f Ku ) − α div ( ∇ u |∇ u | ) = 0 formal Assume K ∗ 1 = 1 in the following (operator scaling) 13 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV EM-Algorithm and TV-Regularization � min Ku − f log Ku + α | u | BV , α > 0 u ∈ BV (Ω) Σ u ≥ 0 Combining nonlocal part (incl. K ) and local regularization functional Convex, but not differentiable Optimality condition for u > 0 � � 1 − K ∗ ( f u Ku ) + α p = 0 , p ∈ ∂ | u | BV 13 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV EM-Algorithm and TV-Regularization Evaluate nonlocal term at u k and subgradient at u k +1 1 − K ∗ ( f ) + α p k +1 = 0 , p k +1 ∈ ∂ | u k +1 | BV Ku k Two-step method 2 = u k K ∗ ( f u k + 1 ) EM step Ku k u k +1 = u k + 1 2 − α u k p k +1 TV step Alternating EM and TV 14 Christoph Brune University of M¨ unster
Introduction EM / Richardson-Lucy EM-TV Methods Nested EM-TV and Analysis Numerical Results Contrast Enhancement by Bregman-EM-TV EM-Algorithm and TV-Regularization Evaluate nonlocal term at u k and subgradient at u k +1 u k +1 − u k K ∗ ( f better: ) + α u k p k +1 = 0 , p k +1 ∈ ∂ | u k +1 | BV Ku k Two-step method 2 = u k K ∗ ( f u k + 1 ) EM step Ku k u k +1 = u k + 1 2 − α u k p k +1 TV step Alternating EM and TV 14 Christoph Brune University of M¨ unster
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